Kinetic Theory Of Gases

The behavior of the gases as stated by various gas laws can be well explained with the help of kinetic
theory of gases. It is explained basing on the molecular nature of matter and the kinetic (faster movement)
movement of gas molecules. Kinetic equation derived with the help of kinetic theory of gases gives a clear
explanation of various gas laws.

Assumptions made from kinetic theory of gases:

As per the kinetic theory of gases, an ideal gas exhibits the below listed properties.

Gases are made up of a huge number of minute particles called molecules. Each of these molecules
is separated from each other by a large distance. Thus, the actual volume occupied by each of these
molecules is very less when compared to the total volume of the gas.

All the molecules of a gas are in constant motion. They move in all directions in straight lines
and collide with each other as well as the wall of the container.

Due to the elastic nature of the gas molecules, there is no loss of energy when the molecules collide.
However, energy is transferred in between the colliding molecules at the time of collision.

Collision of the gas molecules with the walls of the container exerts pressure on the gas molecules.
Greater the number of collisions of the gas molecules with the container greater will be the pressure.

As the distance between the gas molecules is very high, attractive force between the gas molecules and
between the gas molecules and the container is very less. Thus, the molecules move freely.

The kinetic energy of the gas molecules is different and move at different speeds.
However, the average kinetic energy of all the gas molecules is directly proportional to the
absolute temperature. Increase in temperature increases the kinetic energy of the molecules and hence,
they move at greater speed.

Explanation of the above assumptions:

Assumption 1:

When we compress a gas such as oxygen, nitrogen or hydrogen at normal temperature and pressure
(N.T.P), then the volume occupied by the molecules is just 0.014% of the total gas volume and the
rest of it is only an empty space.

Assumption 2:

We observe that in bright light passing through a narrow beam a large number of dust particles
moving in random motion. This is because of the collision between the gas molecules in the air
and the dust particles. During this collision, the gas molecules transfer some of their kinetic
energy to the dust particles which in turn move in zigzag motion termed the Brownian movement.

Assumption 3:

As we know that the gas molecules are in constant motion. This shows that the collisions
between the gas molecules are elastic in nature resulting in the transfer of energy. If
this is not the gas, every collision between the gas molecules results in loss of energy.
Finally, after certain number of collisions, all the gas molecules come to rest. But, this
is not happening. Hence, all the collisions between the gas molecules are elastic in nature
and there is no loss of energy.

Assumption 4:

When the gas molecules are held in a closed container, the rapidly moving gas molecules
collide with the walls of the container. This exerts some pressure on the gas molecules
resulting in the build of pressure within the container. This pressure is termed as
gaseous pressure.

Assumption 5:

When you open the lid of a perfume bottle, you can sense the pleasant smell within seconds
of opening of the lid. This is due to rapid spreading of the gas molecules. This shows that
the gas molecules are held freely without any attractive forces between them.

Assumption 6:

Whenever a gas is heated it evaporates very quickly. This is due to increase in the kinetic energy
of the gas molecules. Thus, the rate at which the gas molecules are moving is determined by the
temperature.

Kinetic Gas Equation:

Now, let us derive a kinetic gas equation basing on the above postulates.

Consider about N molecules of gas each having mass m are enclosed in a container with sides
measuring L cm. From the above postulates we know that gas molecules within a container move
at different speeds. However, the speed of a particular molecule at any moment of time can be
resolved into three components along the three axes of a vessel at right angles to each other.

Let us see how to do this for a single gas molecule having a velocity v. The three velocity
components of the molecule along the three axes X, Y and Z are given as vx, vy,
and vz. The sum of
these velocity components is made equivalent to the velocity v as

V2 = vx2 +, vy2, + vz2

Now, consider a molecule moving along with x-axis with a velocity of vx.

The momentum of the molecule before striking the face A is given as mvx.

Whenever a molecule strikes the walls of a container, it will bounce with same speed but in opposite direction i.e. - vx

Thus, the momentum after striking the surface is given as - mvx

The change in momentum before and after striking is given as mvx- (-mvx) = 2mvx

For the molecule to strike the same face again it has to go to the opposite face and come back. Thus,
the molecule has to travel a distance of 2Lcm. (as the length of each side is Lcm).

The time required for colliding on the same face again is given as

Distance 2L
---------- = ----- sec.
Velocity vx

Number of collisions made by the molecule on the face A per sec is given as

Vx
------ per sec.
2L

As seen above the change in momentum for one collision is given as 2mvx

Though the above equation Is derived for a cubic vessel it is applicable for a vessel of any shape as the
total volume is considered to be made up of a number of small cubes.

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